
As stated in the previous section, for item exposure, maximizing \(g^{\rm item}\)  is equivalent to minimizing the Gini index. The Gini index for \(g^{\rm item}\)  has been routinely used for evaluating inequality in item exposure [1]}, [2]} but there is no algorithm to optimize general trade-offs between user utility and the Gini index of exposure.
[1]} use the Gini index of exposures in the context of dynamic ranking (with the absolute pairwise differences formula (REF )), where their algorithm is shown to asymptotically drive \(g^{\rm item}({v})\)  to 0, equivalent to \(\lambda \rightarrow 1\)  in (REF ). However, their algorithm cannot be used to converge to the optimal rankings for other values of \(\lambda \) . [4]} use as baseline a variant using the standard deviation of exposures instead of absolute pairwise difference because it is easier to optimize (it is smooth except on 0). In contrast, our approach allows for the direct optimization of the welfare function (REF ) with this instantiation of \(g^{\rm item}\)  given by eq. (REF ).
